Theorem eldm1cossres2 39146

Description: Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.)

Assertion

Ref	Expression
eldm1cossres2	⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ [𝑥]𝑅))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑉

Proof of Theorem eldm1cossres2

Step	Hyp	Ref	Expression
1		eldm1cossres 39145	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵))
2		elecALTV 38866	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐵))
3	2	el2v1 38824	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ [𝑥]𝑅 ↔ 𝑥𝑅𝐵))
4	3	rexbidv 3187	. 2 ⊢ (𝐵 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 𝐵 ∈ [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵))
5	1, 4	bitr4d 285	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ [𝑥]𝑅))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2141  ∃wrex 3087  Vcvv 3453   class class class wbr 5108  dom cdm 5661   ↾ cres 5663  [cec 8691   ≀ ccoss 38778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5667  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8695  df-coss 39096

This theorem is referenced by:  eldmqs1cossres  39339